President Trump with President Vladimir V. Putin of Russia at the G-20 summit meeting in Hamburg, Germany, this month. Credit Stephen Crowley/The New York Times

WASHINGTON — The White House acknowledged on Tuesday that President Trump had a second, previously undisclosed, private conversation with President Vladimir V. Putin of Russia this month, raising new questions about their relationship as the cloud of Russia investigations continues to shadow the Trump administration.

The hourlong conversation in Hamburg, Germany, took place at a private dinner of the world leaders at a conference hall on the banks of the Elbe River during the Group of 20 economic summit meeting. It followed a more than two-hour formal meeting earlier in the day between the two presidents that included their foreign ministers and featured a fraught discussion about Moscow’s attempts to interfere in the 2016 elections.

In the earlier meeting, Mr. Trump questioned the Russian president about his role in the American elections, Mr. Putin denied his involvement, and the two men agreed to move beyond the dispute in the interest of finding common ground on other matters, including a limited cease-fire in Syria.

But the intimate dinner conversation, of which there is no official United States government record is the latest to raise eyebrows. Foreign leaders who witnessed it later commented privately on the oddity of an American president flaunting such a close rapport with his Russian counterpart.

“Pretty much everyone at the dinner thought this was really weird, that here is the president of the United States, who clearly wants to display that he has a better relationship personally with President Putin than any of us, or simply doesn’t care,” said Ian Bremmer, president of the Eurasia Group, a Washington-based research and consulting firm, who said he heard directly from attendees. “They were flummoxed, they were confused and they were startled.”

White House officials did not respond to repeated inquiries about the nature or substance of the conversation, but one confirmed that it had occurred without offering any details, and insisted on anonymity because the discussion was private.

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